Nonstationary analysis of cerebral hemodynamics using recursively estimated multiple-input nonlinear models

We present a computational scheme to obtain adaptive non-linear, multiple-input models of the Volterra-Wiener class, by utilizing Laguerre expansions of Volterra kernels in a recursive least-squares formulation. Function expansions have been proven successful in systems identification as they result in a significant reduction of the required free parameters, which is a major limiting factor particularly for nonlinear systems, whereby this number depends exponentially on the nonlinear system order. We apply this scheme in order to obtain adaptive estimates for a two-input model of cerebral hemodynamics, where the two inputs are arterial blood pressure (ABP) and end-tidal CO₂ (P_ETCO2 ) variations and the output is cerebral blood flow velocity (CBFV) variations, by utilizing long-duration (40 min) experimental measurements of spontaneous variations of these signals in healthy humans. Maintenance of a relatively steady cerebral blood flow, despite changes in arterial pressure, is critical in order to meet the high metabolic demands of the brain. This is achieved by the synergistic action of various physiological factors, which may vary over different time-scales and also exhibit nonstationarities. We quantify these nonstationarities for the two main physiological determinants of cerebral blood flow variability (i.e., arterial pressure and arterial CO₂) by considering one- (ABP) and two-input (ABP and P_ETCO2 ) models. The results illustrate the presence of nonstationarities which are frequency-dependent and also that incorporation of P_ETCO2 as an additional input, results in estimates of dynamic pressure autoregulation that are more consistent with respect to time.